Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{-8p^2 - 72p - 144}{-5p^3 + 180p}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ q = \dfrac {-8(p^2 + 9p + 18)} {-5p(p^2 - 36)} $ $ q = \dfrac{8}{5p} \cdot \dfrac{p^2 + 9p + 18}{p^2 - 36} $ Next factor the numerator and denominator. $ q = \dfrac{8}{5p} \cdot \dfrac{(p + 6)(p + 3)}{(p + 6)(p - 6)}$ Assuming $p \neq -6$ , we can cancel the $p + 6$ $ q = \dfrac{8}{5p} \cdot \dfrac{p + 3}{p - 6}$ Therefore: $ q = \dfrac{ 8(p + 3)}{ 5p(p - 6)}$, $p \neq -6$